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Fractions are numbers

Numbers · level 3, lesson 5 of 7

A question to start

Where is \(\tfrac34\) on a number line? Is it between 3 and 4? Between 0 and 1? Somewhere else?

And here is a stranger one: can a fraction be bigger than 1? Is \(\tfrac74\) a real thing, or a mistake? After all, how can you have 7 quarters of a pizza when a pizza only has 4 quarters?

Until now you have met fractions as pieces of something: half a cake, a third of a class. In this lesson you will see fractions as numbers in their own right, with their own places on the number line, in between the whole numbers. That small change of view makes everything about fractions easier.

The idea

The two numbers in a fraction

A fraction like \(\tfrac34\) (say "three quarters", or "three fourths") has two parts:

  • The bottom number, the denominator, tells you how many equal parts the whole is cut into. Here, 4. It names the size of the pieces: quarters. ("Denominator" comes from the Latin word for "namer".)
  • The top number, the numerator, tells you how many of those parts you have. Here, 3. ("Numerator" means "counter".)

A fraction with numerator 1, like \(\tfrac14\), is a unit fraction: one single piece. Every other fraction is made of copies of a unit fraction: \(\tfrac34\) is three copies of \(\tfrac14\), just as 3 apples is three copies of one apple.

The parts must be equal. If a cake is cut into 4 pieces of different sizes, one piece is not a quarter. A quarter is a piece such that exactly 4 of them make the whole.

Fractions live on the number line

Draw a number line with 0 and 1. Cut the gap from 0 to 1 into 4 equal steps. Each step is \(\tfrac14\) long. Starting at 0 and taking 3 of those steps, you land on \(\tfrac34\).

0123 3/45/49/4 = 1 1/4= 2 1/4

So \(\tfrac34\) is a point between 0 and 1, closer to 1. It is a number, just like 3 is a number. It is a distance from 0: three quarters of the way to 1.

This view answers a lot of questions at once:

  • \(\tfrac44\) lands exactly on 1. Four quarters make one whole.
  • \(\tfrac04\) is no steps at all: it is 0.
  • Keep stepping past 1 and you reach \(\tfrac54\), \(\tfrac64\), \(\tfrac74\), \(\tfrac84 = 2\), and so on. There is no reason to stop at 1!

Fractions bigger than one

So yes, \(\tfrac74\) is a real number. It means 7 quarters. You cannot cut 7 quarters from one pizza, but you can from two. On the number line, \(\tfrac74\) is 7 quarter-steps from 0: past 1, and \(\tfrac14\) short of 2.

A fraction whose numerator is as big as its denominator, or bigger, is called an improper fraction. ("Improper" is an old word; there is nothing wrong with these fractions at all. In later years you will use them more than any other kind.)

The same number can also be written as a whole number and a fraction together. \(\tfrac74\) is 1 whole and 3 quarters more, written \(1\tfrac34\) ("one and three quarters"). This is called a mixed number.

111/3 7 thirds = 7/3 = 2 1/3

Changing between the two ways of writing

Improper to mixed: divide. How many wholes are in \(\tfrac{17}{5}\)? Every 5 fifths make one whole, so ask how many 5s fit in 17: \(17 \div 5 = 3\) remainder 2. So 3 wholes and 2 fifths left over:

\[\frac{17}{5} = 3\tfrac25\]

Mixed to improper: multiply and add. How many fifths are in \(3\tfrac25\)? Each whole is 5 fifths, so 3 wholes are \(3 \times 5 = 15\) fifths. Add the 2 extra fifths: 17 fifths.

\[3\tfrac25 = \frac{3 \times 5 + 2}{5} = \frac{17}{5}\]

These two moves undo each other, just as division and multiplication do.

A fraction is also a division

Here is the most surprising fact in this lesson. Share 3 pizzas among 4 people. How much does each person get?

Cut every pizza into quarters. That makes 12 quarters. Shared among 4 people, each gets 3 quarters. So

\[3 \div 4 = \frac34\]

This always works: \(\tfrac{a}{b}\) is the same as \(a \div b\). The fraction bar is a division sign. That is why \(\tfrac{12}{4} = 3\) (12 shared by 4) and why \(\tfrac{7}{3}\), which is \(7 \div 3 = 2\) r 1, is \(2\tfrac13\): the remainder from the last lesson has become the fraction part.

So the leftover 1 pizza in "7 pizzas shared by 3" is not a problem anymore. It is cut into 3 parts, and each person gets \(\tfrac13\) of it.

Wholes can be anything

A fraction is always a fraction of something. Half of a small pizza is less pizza than half of a large one. On the number line, the whole is the length from 0 to 1. In a story, ask: "a third of what?" A third of a class of 24 is 8 children; a third of a sol (a Mars day, about 24 hours and 40 minutes) is a bit more than 8 hours.

Worked examples

1. Place a fraction. Where is \(\tfrac25\) on the number line?

Cut the gap from 0 to 1 into 5 equal steps (fifths). Take 2 steps from 0. It is a little less than halfway to 1.

2. Improper to mixed. Write \(\tfrac{23}{4}\) as a mixed number.

\(23 \div 4 = 5\) r 3 (because \(4 \times 5 = 20\)). So \(\tfrac{23}{4} = 5\tfrac34\). Check: \(5 \times 4 + 3 = 23\). ✓

3. Mixed to improper. Write \(4\tfrac23\) as an improper fraction.

4 wholes are \(4 \times 3 = 12\) thirds. Plus 2 thirds is 14 thirds: \(4\tfrac23 = \tfrac{14}{3}\).

4. Between which whole numbers? Between which two whole numbers is \(\tfrac{30}{7}\)?

\(7 \times 4 = 28\) and \(7 \times 5 = 35\). So 30 sevenths is more than 4 wholes but less than 5: it is between 4 and 5. Exactly, \(\tfrac{30}{7} = 4\tfrac27\).

5. Fraction as division. 5 friends share 2 big chocolate bars equally. How much does each get?

\(2 \div 5 = \tfrac25\) of a bar. Check: cut each bar into 5 pieces, making 10 pieces. 10 pieces among 5 friends is 2 pieces each, and each piece is a fifth of a bar. ✓

Try it

  1. In \(\tfrac58\), what does the 8 tell you? What does the 5 tell you?
  2. Draw a number line from 0 to 2 cut into thirds. Mark \(\tfrac13\), \(\tfrac23\), \(\tfrac43\) and \(\tfrac53\).
  3. Write as mixed numbers: (a) \(\tfrac94\) (b) \(\tfrac{11}{3}\) (c) \(\tfrac{20}{6}\) (d) \(\tfrac{16}{8}\).
  4. Write as improper fractions: (a) \(2\tfrac12\) (b) \(3\tfrac34\) (c) \(1\tfrac{5}{8}\).
  5. Which whole number is \(\tfrac{19}{5}\) closest to?
  6. 3 cakes are shared equally by 8 people. How much cake does each person get?
  7. How many quarters are in 6? How many fifths are in \(2\tfrac35\)?
  8. Harder: Mia says "\(\tfrac{6}{5}\) is impossible because you can't take 6 pieces out of 5." Explain to Mia what \(\tfrac65\) means, with a picture or a number line.
Answers
  1. The 8 says the whole is cut into 8 equal parts (eighths). The 5 says we have 5 of those parts.
  2. Each third is one step. \(\tfrac13\) and \(\tfrac23\) are between 0 and 1; \(\tfrac33\) is exactly 1; \(\tfrac43\) is one step past 1 and \(\tfrac53\) two steps past 1; \(\tfrac63\) is 2.
  3. (a) \(9 \div 4 = 2\) r 1: \(2\tfrac14\). (b) \(11 \div 3 = 3\) r 2: \(3\tfrac23\). (c) \(20 \div 6 = 3\) r 2: \(3\tfrac26\), which is also \(3\tfrac13\) (you will see why next lesson). (d) \(16 \div 8 = 2\) r 0: exactly 2.
  4. (a) \(\tfrac{2 \times 2 + 1}{2} = \tfrac52\). (b) \(\tfrac{3 \times 4 + 3}{4} = \tfrac{15}{4}\). (c) \(\tfrac{1 \times 8 + 5}{8} = \tfrac{13}{8}\).
  5. \(\tfrac{19}{5} = 3\tfrac45\), which is only \(\tfrac15\) away from 4. Closest to 4.
  6. \(3 \div 8 = \tfrac38\) of a cake each.
  7. Each whole has 4 quarters, so 6 has \(6 \times 4 = 24\) quarters. \(2\tfrac35\) is \(2 \times 5 + 3 = 13\) fifths.
  8. \(\tfrac65\) means 6 pieces, each one fifth of a whole. You need more than one whole to get them: take one whole bar (5 fifths) and one fifth of another bar. On the number line it is 6 fifth-steps from 0: one step past 1. So \(\tfrac65 = 1\tfrac15\).

Common mistake

Thinking the bigger denominator means the bigger fraction. \(\tfrac18\) is smaller than \(\tfrac14\), not bigger. Cutting a whole into more pieces makes each piece smaller. 8 people sharing a pizza get less each than 4 people do.

Unequal parts. A shape cut into 4 pieces of different sizes does not show quarters. Before you name a fraction, check that the parts are equal.

Where it leads